Control of non-linear systems

A similar block diagram may be constructed for the temperature process. Actuator .... controllers will be designed for these three models; very simple tuning.
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1. Case study: Investigation of process nonlinearities - Instrutek VVS-400 heating and ventilation process

Control of non-linear systems Structure of discussion • Case study: Investigation of process nonlinearities - Instrutek VVS-400 heating and ventilation process • Gain scheduling • Control loop characterisers • Tutorial questions

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Flow control system block diagram – local controller used command value SV (0-100%)

controller output MV (0-100%)

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Controller

Fan voltage

Signal conditioning

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controlled variable PV (0-100%)

Signal conditioning

Pressure

Fan speed Actuator (Fan)

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Flow “Process” – Computer control • The process is linked via a data acquisition package, to MATLAB. • The controller output is sent out through a data acquisition card, with a range of 0 to 5V. • The output of the orifice plate is converted by the signal conditioning circuitry of the Instrutek rig to a voltage signal, which in turn is input to the PC using the data acquisition card. • The concept is summarised in the block diagram. • The block diagram represents the effective (dynamic) relationship between the manipulated variable (i.e. the controller output signal) and the controlled variable (i.e. the process output signal). The controller will, in general, be designed based on this relationship.

Flow Process

Orifice plate

A similar block diagram may be constructed for the temperature process. DAC

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Fan

Process

Orifice plate

Similar issues arise for the temperature process.

Signal conditioning

DAC

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Flow process static tests

Interfacing Hardware with MATLAB/Humusoft

Open loop static tests were carried out, at controller output signals (proportional to fan speed) of 0%, 5%, 10% …. 100% of maximum, and the process output signal (proportional to flow) was recorded. The load vane was fully open. Separate tests were carried out when the controller output signals increased from 0 to 100%, and controller output signals decreased from 100% to 0%. The plot shown results; the static characteristic is clearly non-linear.

• Deadband • Bias • Gain is greater for larger inputs • Small hysteresis 5

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Flow process

Flow process dynamic tests • Open loop step tests, using small steps over the full range of fan speed were carried out, to obtain dynamic models for the flow process at different operating points. • The load vane was fully open. • A tangent and point method was used to approximate the process as a first order lag plus time delay (FOLPD) model.

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The tangent and point method was applied to each plot to determine the model parameters. Typical result:

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Flow process – overall dynamic results

Open loop static tests were carried out, at controller output signals (proportional to heater setting) of 0%, 10%, 20% …. 100% of maximum, and the process output signal (proportional to temperature) was recorded. The load vane was fully open. Separate tests were carried out when the controller output signals increased from 0 to 100%, and controller output signals decreased from 100% to 0%. In addition, these tests were carried out at three fan speed settings (30%, 50%, and 70% of maximum). The plot shown results; the static characteristic is clearly non-linear.

The results also show that the flow process is non-linear; A summary of the average results is as follows:

G m (s) =

0.45e −0.98s 1 + 2.70s

Fan speed setting < 55% of maximum

1.08e −1.08s G m (s) = 1 + 1.93s

Fan speed setting = [55%,75%] of maximum

1.76e −0.93s G m (s) = 1 + 1.45s

Fan speed setting > 75% of maximum

In fact, model parameters vary at each fan setting taken, and vary too if fan speed is increased or decreased. Models were not determined at a range of heater settings (why ?)

Temperature process static tests

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Temperature process dynamic tests

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Temperature process – overall dynamic results

• Open loop step tests, using small steps over the full range of heater setting were carried out. • In addition, these tests were carried out at three fan speed settings (30%, 50%, and 70% of maximum) • The load vane was fully open. • The alternative tangent and point method was used to approximate the process as a first order lag plus time delay (FOLPD) model. • Typical result:

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The results show that the temperature process is non-linear. A summary of the average results is as follows:

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2. Gain scheduling

Clearly, the temperature process is nonlinear. In addition, model parameters vary at each heater setting taken, and vary too if heater setting is increased or decreased. Furthermore, the measurement problem is greater ! (i.e. signal/noise < 5).

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Gain scheduler design

Gain scheduler design The gain scheduler may be implemented in SIMULINK software (though the details are involved). Instead, consider the regulator response of the flow system (at a low flow setting) for (1) the gain scheduled controller design and (2) a fixed PI controller design. Trade-off: increased performance vs. increased design complexity.

For the flow process (based on the experimental work):

0.45e −0.98s 1 + 2.70s 1.08e −1.08s G m (s) = 1 + 1.93s

G m (s) =

1.76e −0.93s G m (s) = 1 + 1.45s

With this strategy, the parameters of the controller are adjusted based on knowledge of the plant at different operating points. For the Instrutek VVS-400 process discussed, both the flow and temperature processes are known at different operating points, based on the extensive experimental work performed; thus, logically, controllers may be designed based on the process models identified. There is a need to change controller settings as the operating point of the process changes; thus, there is a need for a scheduling variable to signal to the controller such a change in the operating point. For instance, for the VVS-400 temperature control process, there will be two scheduling variables; the temperature measured (by the temperature sensor) and the flow measured (by the orifice plate). The VVS-400 process will be considered as a case study. 14

Fan speed setting < 55% of maximum

Fan speed setting = [55%,75%] of maximum

Fan speed setting > 75% of maximum

The scheduling variable is flow (as measured by the orifice plate) which is nonlinearly related to fan speed setting (see flow process static tests). PI controllers will be designed for these three models; very simple tuning rules will be used. For instance, for good regulator tuning, the following tuning rule is appropriate: 0.986 0.707

Kc =

0.984  Tm    K m  τ m 

Ti =

Tm  τ m    0.608  Tm 

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Level control – spherical tank

Application 2: Level control of a spherical tank

Thus, if the PID controller parameters are fixed, the level controller becomes very oscillatory at a higher process output, due to the high value of process gain. If the level signal is used as the scheduling variable, then a number of sets of controller parameters may be specified for different levels (depending on the accuracy required). Typical responses are shown.

The process gain of the tank varies with level (i.e. the increase in level as a result of the addition of a fixed quantity of water will be smaller when the level is at 50% of its maximum value, compared to when the level is at 80% of its maximum value). Reference: Hang, C.C. et al. (1993). Adaptive control, Instrument Society of America, Chapter 1 17

3. Control loop characterisers As has been seen, one method to control a non-linear process is to obtain a linear process model, perhaps at a number of operating points, and either choose a single set of PID controller parameters for all operating conditions, or use gain scheduling with a look-up table to set up relevant PID controller parameters. Source: Shinskey, F.G. (1999). “Characterisers for control loops”, Expertune document, May. An alternative is to apply a curve characteriser (for processes with a static non-linearity). A pH control system gives an interesting example. A typical titration curve for industrial wastewater is shown. This is a static response, so this curve shows how process gain varies with millilitres of caustic (base) added. 19

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Control loop characterisers The desired pH is normally 7; the process gain is approximately 75 for a pH range of 5 to 9. The controller gain must be adjusted so that the closed loop system performs satisfactorily in this range. Figure 2 shows the response of the pH control loop, with a fixed PI controller, following a step decrease in acid load (disturbance). The PI controller is tuned for light damping at SP. However, the controller gain is low, so that recovery of PV from the disturbance is very slow. For small disturbances, the response would be expected to be better. 20

Control loop characterisers

Control loop characterisers

The closed loop control system may be linearised by placing a complementary non-linear function in series with the process (a characteriser); in this example, the characteriser is the inverse of the pH titration curve.

Now, as a result of the linearisation, the controller gain may be increased (as the overall process gain is reduced). The response of the closed loop system, following a step decrease in acid load is as follows:

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4. Tutorial question

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Tutorial question - solution

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Question and answer

Tutorial question - solution

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